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Long time behaviour of viscous scalar conservation laws
Anne-Laure Dalibard
To cite this version:
Anne-Laure Dalibard. Long time behaviour of viscous scalar conservation laws. Indiana Uni- versity Mathematics Journal, Indiana University Mathematics Journal, 2010, 59 (1), pp.257-300.
�10.1512/iumj.2010.59.3874�. �hal-00345324�
CONSERVATION LAWS
ANNE-LAURE DALIBARD
Abstract. This paper is concerned with the stability of stationary solutions of the conservation law∂tu+ divyA(y, u)−∆yu = 0, where the flux A is periodic with respect to its first variable. Essentially two kinds of asymptotic behaviours are studied here: the case when the equation is set on R, and the case when it is endowed with periodic boundary conditions. In the whole space case, we first prove the existence of viscous stationary shocks - also called standing shocks - which connect two different periodic stationary solutions to one another. We prove that standing shocks are stable inL1, provided the initial disturbance satisfies some appropriate boundedness conditions. We also extend this result to arbitrary initial data, but with some restrictions on the fluxA. In the periodic case, we prove that periodic stationary solutions are always stable. The proof of this result relies on the derivation of uniformL∞ bounds on the solution of the conservation law, and on sub- and super-solution techniques.
Keywords. Viscous shocks; shock stability; viscous scalar conservation laws.
AMS subject classifications. 35B35, 35B40, 76L05.
1. Introduction
This paper is devoted to the analysis of the long-time behaviour of the solution u∈ C([0,∞), L1loc(Q))∩L∞loc([0,∞), L∞(Q)) of the equation
(1) ∂tu+ divyA(y, u)−∆yu= 0, t >0, y∈Q, u|t=0=u0∈L∞(Q).
Above, Q denotes eitherR or TN, the N-dimensional torus (TN =RN/[0,1)N), andA∈Wloc1,∞(TN×R)N is anN-dimensional flux (withN = 1 whenQ=R).
Heuristically, it can be expected that the parabolicity of equation (1) may yield some compactness on the trajectory{u(t)}t≥0.Hence, it is legitimate to conjecture that the family u(t) will converge ast → ∞ towards a stationary solution of (1).
Such a result was proved when Q = TN by the author in [6] for a certain class of initial conditions, namely when u0 is bounded from above and below by two stationary solutions of (1). Such an assumption is in fact classical in the framework of conservation laws which admit a comparison principle: we refer for instance to [2], where the authors study the long time behaviour of the fast diffusion equation, and assume that the initial data is bounded by two Barenblatt profiles. The same kind of assumption was made in the context of travelling waves by Stanley Osher and James Ralston in [17]; let us also mention the review paper by Denis Serre [19], which is devoted to the stability of shock profiles of scalar conservation laws, and in which the author assumes at first that the initial data is bounded from above
1
and below by shifted shock profiles. Nonetheless, in [9] (see also [18, 19]), Heinrich Freist¨uhler and Denis Serre remove this hypothesis, and prove that shock stability holds under a mere L1assumption on the initial data.
The goal of this paper is to extend the result of [6] to arbitrary initial data, that is, to prove that solutions of (1) converge towards a stationary solution for any initial data u0 ∈ L∞(TN). We also tackle similar issues on the stability of stationary shock profiles in dimension one, when the equation is set on the whole space case (Q = R). Thus, a large part of the paper is devoted to the proof of the existence of shock profiles, and to the analysis of their properties. We will see that the question of shock stability reduces in fact to the stability of periodic stationary solutions of (1) inL1(R). Unfortunately, we have been unable to prove that periodic stationary solutions of (1) are stable in L1(R) for arbitrary fluxes.
Hence we have left this issue open, and we hope to come back to it in a future paper.
The proof of stability in the periodic setting relies strongly on the derivation of uniformL∞bounds on the family{u(t)}t≥0. In the whole space case, the first step of the analysis is to prove the property for initial data which are bounded from above and below by viscous shocks; in fact, this result is similar to the one proved in [6], and uses arguments from dynamical systems theory, following an idea by S. Osher and J. Ralston [17] (see also [19, 1]). But the derivation of uniformL∞ bounds is not sufficient to obtain a general stability result in the whole space case.
Thus the idea is to use existing results on the long time behaviour of transport equations, which were obtained by Adrien Blanchet, Jean Dolbeault and Michal Kowalczyk in [3], and to apply those in the present context.
Throughout the paper, we use the following notation: ifv∈L1(TN), hvi=
Z
TN
v.
We denote byL10(Q) the set of intergrable functions with zero mass:
L10(Q) :={u∈L1(Q), Z
Q
u= 0}.
Following [13], forα∈(0,1), we define, if I is an interval in (0,∞) and Ω is a domain inRN,
Hα2,α(I×Ω) ={u∈ C( ¯I×Ω),¯ kukHα/2,α(I×Ω)<∞}, where
kukHα2,α
(I×Ω) := max
(t,x)∈Iׯ Ω¯|u(t, x)|
+ sup
(x,t)∈I×Ω, (x′,t′)∈I×Ω,
|t−t′|≤ρ
|u(t, x)−u(t′, x)|
|t−t′|α/2 + sup
(x,t)∈I×Ω, (x′,t′)∈I×Ω,
|x−x′|≤ρ
|u(t, x)−u(t, x′)|
|x−x′|α ;
above,ρis any positive number. We also set Cα(Ω) :=
(
u∈ C( ¯Ω), sup
(x,x′)∈Ω2
|u(x)−u(x′)|
|x−x′|α <+∞ )
. Eventually, forf ∈L1loc(R),h∈R, we setτhf =f(·+h).
2. Main results
Before stating our main results, we recall general features of equation (1), to- gether with some facts related to the stationary solutions of this equation.
In the rest of this paper, we denote byStthe semi-group associated with equation (1). Notice that St is always well-defined on L∞(Q), thanks to the papers by Kruˇzkov [11, 12]. Moreover, we recall that the following properties hold true (these are called theCo-propertiesin [19]):
• Comparison: ifa, b∈L∞(Q) are such thata≤b, then Sta≤Stbfor all t≥0.
• Contraction: ifa, b∈L∞(Q) are such thata−b∈L1(Q), thenSta−Stb∈ L1(Q) for allt≥0 and
kSta−StbkL1 ≤ ka−bkL1 ∀t≥0.
• Conservation: ifa, b∈L∞(Q) are such thata−b ∈L1(Q), then Sta− Stb∈L1(Q) for allt≥0 and
Z
Q
(Sta−Stb) = Z
Q
(a−b) ∀t≥0.
Thanks to the Contraction property, the semi-groupStcan be extended onL∞(Q)+
L1(Q). The so-called “Constant property” in [19] is not true in the present setting, since the fluxAdoes not commute with translations. In other words, constants are not stationary solutions of equation (1) in general. The existence of periodic (in space) stationary solutions of (1) was proved by the author in [5], and we recall the corresponding result below:
Proposition 2.1. Let A ∈ Wloc1,∞(TN ×R)N. Assume that there exist C0 > 0, m∈[0,∞),n∈[0,N+2N−2)when N≥3, such that for all(y, p)∈TN ×R
|∂pAi(y, p)| ≤C0(1 +|p|m) ∀ 1≤i≤N, (2)
|divyA(y, p)| ≤C0(1 +|p|n). (3)
Assume as well that one of the following conditions holds:
m= 0or0≤n <1 or
n <min
N+ 2 N ,2
and∃p0∈R, divyA(·, p0) = 0
. (4)
Then for all p ∈ R, there exists a unique solution v(·, p) ∈ Hper1 (TN) of the equation
(5) −∆yv(y, p) + divyA(y, v(y, p)) = 0, hv(·, p)i=p.
The family (v(·, p))p∈R satisfies the following properties:
(i) Regularity estimates: For all p∈R, v(·, p)belongs to Wper2,q(TN) for all 1<
q <+∞ and additionally
(6) ∀R >0 ∃CR>0 ∀p∈[−R, R] ||v(·, p)||W2,q(TN)≤CR. (ii) Growth property: ifp > p′, then
v(y, p)> v(y, p′) ∀y∈TN.
(iii) Behaviour at infinity: assume that
(7) sup
v∈Rk∂vA(·, v)kL∞(TN)<+∞. Then
p→−∞lim sup
y∈TN
v(y, p) =−∞, lim
p→+∞ inf
y∈TNv(y, p) = +∞.
2.1. A priori bounds for solutions of scalar conservation laws. Our first result is concerned with the derivation ofa priori bounds inL∞which are uniform in time. Notice that such a result is not trivial in general: in the homogeneous case, that is, when the fluxA does not depend on the space variablex, this result follows from the comparison principle stated earlier. However, in the present case, this argument does not hold, since constants are not stationary solutions of (1).
Of course, if there exists a constantC such thatu0≤v(·, C), then the comparison principle entails that Stu0 ≤ v(·, C). Hence, the derivation of a priori bounds is easy when the initial data is bounded from above and below by solutions of equation (5). Consequently, the goal of this paragraph is to present similar results when the initial data does not satisfy such an assumption.
Proposition 2.2. Assume that the flux Asatisfies the assumptions of Proposition 2.1. Assume also that for allK >0, there exists a positive constantCK, such that for allv∈[−K, K], for all w∈R,
(8) |divyA(y, v+w)−divyA(y, v)| ≤CK(|w|+|w|n),
|∂vA(y, v+w)−∂vA(y, v)| ≤CK(|w|+|w|n), wheren <(N+ 2)/N.
Let u0 ∈ L∞(Q), and assume that there exists a stationary solution U0 ∈ W1,∞(Q)of (1)such that u0∈U0+L1(Q).
Then
sup
t≥0kStu0kL∞(Q)<+∞.
Notice that in the above proposition, we do not assume that the stationary solutionU0 is periodic. Thus U0 is not necessarily a solution of equation (5), and may be, for instance, a viscous shock profile (see Proposition 2.4 below). In the periodic case, any functionu0 ∈L∞ is such that u0−v(·,0) ∈L1(TN), and thus the result holds for all functions inL∞.
2.2. Stability of stationary periodic solutions in the periodic case. The derivation of uniforma priori bounds for the solutions of equation (1) allows us to extend the stability results proved in [6] to general classes of initial data. Let us first recall the stability result of [6]:
Proposition 2.3. Assume that the flux Asatisfies the assumptions of Proposition 2.1. Letu0∈L∞(TN)such that there existsβ1, β2∈Rsuch that
(9) v(·, β1)≤u0≤v(·, β2).
Then ast→ ∞
Stu0→v(·,hu0i) inL∞(TN).
It was also proved in [6] that under additional regularity assumptions on the flux A, the speed of convergence is exponential, due to a spectral gap result.
We now remove assumption (9) thanks to Proposition 2.2:
Theorem 2.1. Assume that the flux A satisfies the assumptions of Proposition 2.1, together with (8). Then for all u0∈L∞(TN), ast→ ∞,
Stu0→v(·,hu0i) inL∞(TN).
The proof of this result relies mainly on Proposition 2.2 and on sub- and super- solution methods based on the Comparison principle. Once again, it can be proved that the speed of convergence is exponential, provided the flux A is sufficiently smooth. For details regarding that point, we refer to [6].
2.3. Existence of viscous shocks. We now consider equation (1) set in Q=R. Our goal here is to prove the stability of a special class of stationary solutions, called
“standing shocks”. By analogy with the definition in [19] of shocks in homogeneous conservation laws, astanding shock is a stationary solutionU of equation (1) which is asymptotic to solutions of equation (5) at infinity, namely
∃(pl, pr)∈R2, lim
y→−∞(U(y)−v(y, pl)) = 0, lim
y→+∞(U(y)−v(y, pr)) = 0.
Because of the spatial dependence of the fluxA, it does not seem to be possible to restrict the study of general shocks to standing shocks. For that matter, we wish to emphasize that the definition of a viscous shock with non-zero speed should not be exactly the same as in [19]; indeed, it can be easily checked that if
u(t, x) =U(x−st)
is a solution of (1), then s = 0 necessarily. Thus, for s 6= 0, a shock profile is a solution of (1) of the form
u(t, x) =U(t, x−st),
where for all t, U(t) is asymptotic to solutions of equation (5) at infinity. This is related (although not equivalent to) the definition of traveling pulsating fronts, see for instance the paper of Xue Xin [20] The existence of non-stationary shock profiles and their stability is beyond the scope of this paper, and thus, we will focus on standing shocks from now on, sometimes omitting the word “standing”.
Our first result is concerned with the existence of viscous shocks.
Proposition 2.4(Existence of stationary shock profiles). Assume that there exists p−, p+∈Rsuch that p−< p+ and
(10) A(p¯ +) = ¯A(p−) :=α,
and define v±:=v(·, p±).
Let u0∈Rsuch that
v−(0)< u0< v+(0),
and letu:I→Rbe the maximal solution of the differential equation u′(x) =A(x, u(x))−α,
(11)
u|x=0=u0. (12)
Then usatisfies the following properties:
(i) The functionuis a global solution of (11); in other words, I=R.
(ii) For allx∈R,
v−(x)≤u(x)≤v+(x);
(iii) There exist ql, qr∈Rsuch thatql6=qr,A(q¯ l) = ¯A(qr) =α, and
x→−∞lim (u(x)−v(x, ql)) = 0, lim
x→+∞(u(x)−v(x, qr)) = 0.
As a consequence, the solution u of (11)-(12)is a stationary viscous shock profile of equation (1).
Remark 1. (i) Assumption (10) is the analogue of the Rankine-Hugoniot con- dition for homogeneous conservation laws. It is in fact a necessary condition, as demonstrated in Lemma 3.1 below.
(ii) In general, the asymptotic statesv(ql), v(qr) are different fromv(p+), v(p−).
Proposition 2.4 only ensures that
A(q¯ l) = ¯A(qr) = ¯A(p±).
However, under an additional condition of Oleinik type, the asymptotic states can be identified:
Corollary 2.1. Assume that the hypotheses of Proposition 2.4 are satisfied, and that the flux A¯ is such that
(13) ∀p∈(p−, p+), A(p)¯ 6=α.
Then
{ql, qr}={p+, p−}.
2.4. Stability of stationary shocks in the whole space case. We are now ready to state results on shock stability for equation (1). Our first result is the analogue of Proposition 2.3: indeed, Theorem 2.2 below states thatStu0converges towards a viscous shock, provided u0 is bounded from above and below by the asymptotic states of the shock. In view of Theorem 2.1, it is natural to expect that this result remains true for arbitrary initial data. Unfortunately, we have not been able to prove this result in complete generality: we prove that stationary shocks are stable in L1 provided stability holds (in L1(R)) for solutions of equation (5). We also give explicit examples of fluxes for which the stability of shocks and periodic stationary solutions can be established.
Theorem 2.2. Assume that the flux A satisfies the assumptions of Proposition 2.1. Let pl, pr ∈ R such that pl 6=pr and A(p¯ r) = ¯A(pl) =: α, and assume that A, p¯ l, pr satisfy Oleinik’s condition (13).
LetU be a shock profile connectingv(pl)tov(pr). Letu0∈U+L1(R)such that for almost everyx∈R,
(14) v(x,min(pl, pr))≤u0(x)≤v(x,max(pl, pr)).
Then there exists a shock profile V connecting v(pl) to v(pr) and such that u ∈ V +L10(R).Moreover,
t→∞lim kStu0−VkL1(R)= 0.
As outlined before, hypothesis (14) should be compared with assumption (9).
Thus, the next step would be to prove that stability holds even when (14) is false.
In fact, we are only able to prove the following:
Proposition 2.5. Assume that the flux Asatisfies the assumptions of Proposition 2.2. Letpl, pr∈Rsuch thatpl6=pr,A(p¯ r) = ¯A(pl), and such that (13)is satisfied.
Assume that the following assertion is true:
(H)Forp∈ {pl, pr}, for allu0∈v(·, p) +L10(R), lim
t→∞kStu0−v(·, p)kL1(R)= 0.
LetU be a shock profile connectingv(pl)tov(pr), and letu0∈U+L1(R). Then there exists a shock profileV connectingv(pl)tov(pr)and such thatu∈V+L10(R).
Moreover,
t→∞lim kStu0−VkL1(R)= 0.
Remark 2. • As we will see in Section 7, hypothesis(H)can be relaxed into (H’)Forp∈ {pl, pr}, there existsδ >0 such that for allu0∈v(·, p) +L10(R) ,
ku0−v(p)k1≤δ⇒ lim
t→∞kStu0−v(·, p)kL1(R)= 0.
• In Section 7, we will prove the following result: for allp∈R, there existsδ > 0 such that ifu0∈v(·, p) +L10(R) satisfiesku0−v(p)k1≤δ, then
t→∞lim kStu0−v(·, p)kL∞(R)= 0.
Hence, in this case, for all ε >0 there existstε≥0 such that Stεu0∈v(·, p) +L10 and
v(·, p−ε)≤u0≤v(·, p+ε).
Consequently, hypothesis(H)can also be relaxed into
(H”)For allp∈R, there existsδ >0 such that for allu0∈v(·, p) +L10(R) v(·, p−δ)≤u0≤v(·, p+δ)⇒ lim
t→∞kStu0−v(·, p)kL1(R)= 0.
To sum up, denoting by (C) the conclusion of Proposition 2.5 (that is, shock stability), we have roughly
(H)⇒(H”)⇒(H’)⇒(C).
We now give an example when it is known that(H)is true. This example relies on the analysis performed in the linear case by A. Blanchet, J. Dolbeault and M.
Kowalczyk (see [3]).
Proposition 2.6. Assume that the flux A satisfies the hypotheses of Proposition 2.2, and let p∈R. Assume that A is linear in a neighbourhood of v(·, p), i.e.
∃b∈ C1(TN), ∃η >0, ∀ξ∈(−η, η), A(y, v(y, p) +ξ) =A(y, v(y, p)) +b(y)ξ.
Then, provided a technical assumption on the moments of Stu0 is satisfied (see (42)), there existsδ >0 such that for allu0∈v(·, p) +L10,
ku0−v(·, p)k1≤δ⇒ lim
t→∞kStu0−v(p)k1= 0.
The assumption (42) is a little technical to state at this stage, and is inherited from the analysis in [3]. However, as explained in [3], this hypothesis is expected to be satisfied for a large class of initial data, so that in fact (42) is not a restriction.
This allows us to give an explicit case of flux for which shock stability holds.
Corollary 2.2. Assume that the fluxA is given by A(y, v) =V(y) +f(v),
where V ∈ C2(TN)and f ∈ C2(R) is a convex function which is linear at infinity, i.e.
∃(a−, a+)∈(0,∞)2, ∃A >0
f(v) =a+v if v > A, f(v) =−a−v if v <−A.
Then the following properties hold:
(i) For all α > 0 large enough, there exist (pl, pr) ∈R2 such that pl 6=pr and A(p¯ l) = ¯A(pr) = α, and there exists a shock profile U connecting v(pl) to v(pr).
(ii) Letu0∈U+L1(R). Then there exists a shock profileV such thatu∈V+L10. Moreover, if (42)holds for any v0 ∈ v(p±) +L10(R), then limt→∞kStu0− Vk1= 0.
The plan of the paper is the following: given the similarity between the state- ments for periodic solutions whenQ = TN, and stationary shocks when Q =R, we first prove the existence of standing shocks (i.e. Proposition 2.4) and the shock stability result under boundedness conditions on the initial data (i.e. Theorem 2.2) in sections 3 and 4 respectively. At this stage, we are able to treat both models simultaneously, and thus we prove Proposition 2.2 in Section 5. Section 6 is devoted to the proof of Theorem 2.1, and at last, we give in Section 7 further results on shock stability, including the proofs of Propositions 2.5 and 2.6.
3. Existence of one dimensional stationary viscous shocks This section is devoted to the proof of Proposition 2.4, together with a number of results related to viscous shocks which will be useful in the proof of Theorem 2.2. These auxiliary results (monotonicity of shock profiles, integrability of the difference between two shock profiles, etc.) can be found in paragraph 3.3. We also give in paragraph 3.4 a few explicit examples in the case when the fluxAis convex.
We begin with some comments on assumption (10).
3.1. Analysis of necessary conditions.
Lemma 3.1. Let ql, qr∈R, and letu∈W1,∞(R)be such that u(x)−v(x, ql)→0 asx→ −∞, u(x)−v(x, qr)→0 asx→+∞,
−u′′+ d
dx(A(x, u(x))) = 0.
Then A(q¯ r) = ¯A(ql) =:α,andusatisfies
−u′+A(x, u(x)) =α.
Proof. We deduce from the differential equation that there exists a constantCsuch that
−u′+A(x, u) =C,
and the goal is to prove that ¯A(qr) =C= ¯A(ql). We recall first that for allp∈R, v(·, p) is a solution of
−v′(x, p) +A(x, v(x, p)) = ¯A(p).
Indeed, integrating (5) onR, we infer that for allp∈Rthere exists a constantCp
such that
∀x∈R, −v′(x, p) +A(x, v(x, p)) =Cp.
Taking the average of the above equality over a period, we deduce thatCp= ¯A(p).
As a consequence, we have (15) − d
dx(u(x)−v(x, qr)) + [A(x, u(x))−A(x, v(x, qr))] =C−A(q¯ r) Now, letδ >0 arbitrary. There existsxr>0 such that
x≥xr⇒(|u(x)−v(x, qr)| ≤δ, |A(x, u(x))−A(x, v(x, qr))| ≤δ). Integrating (15) on the interval [xr, xr+ 1], we deduce that
|C−A(q¯ r)| ≤3δ.
Since the above inequality is true for allδ >0, we infer thatC= ¯A(qr). The other equality is treated similarly.
Remark 3. Notice that couples (pl, pr) such that pl 6=pr and ¯A(pl) = ¯A(pr) do not always exist. Indeed, consider the case of a linear flux A(x, v) = a(x)v, with a∈ C1(T). Then, for allp∈ R, we havev(x, p) =pm(x), wherem is the unique probability measure onTsatisfying
−m′′(x) + d
dx(a(x)m(x)) = 0, x∈T.
The positivity ofmis a consequence of the Krein-Rutman Theorem; we refer to [5]
for more details.
Therefore, for allp∈R,
A(p) =¯ hav(·, p)i=phami.
Hence, as long as hami 6= 0, ¯A(p) 6= ¯A(q) for all p, q ∈ R such that p 6= q. In particular, ifais a non-zero constant, assumption (10) is never satisfied.
3.2. Proof of Proposition 2.4. We begin with the a priori bound (ii), from which we deduce thatuis a global solution.
The inequality(ii)follows directly from classical results in differential equations;
indeed, assume that there existsx0∈I such that u(x0)≥v+(x0);
sinceu(0)< v+(0), there existsx1∈[0, x0] such thatu(x1) =v+(x1). Butuand v+ are solutions of the same differential equation, whence the Cauchy-Lipschitz Theorem implies thatu=v+, which is false. Thus
u(x)< v+(x) ∀x∈I.
The lower bound is proved in the same way.
As a consequence, we deduce thaturemains bounded on its (maximal) interval of existenceI. Using once again the Cauchy-Lipschitz Theorem, we infer thatI=R, and thusuis a global solution.
We now tackle the core of Proposition 2.4. First, since the fluxA isT-periodic, the function u(·+ 1) is also a solution of equation (11). Hence the function x7→
u(x+ 1)−u(x) keeps a constant sign on R, which entails in particular that for
allx∈R, the sequences (u(x±k))k∈N are monotonous. Consider for instance the sequence of functions
uk:x∈[0,1]7→u(x+k).
According to the above remarks, the sequence (uk) is monotonous and bounded in L∞; hence for allx ∈[0,1], uk(x) has a finite limit, which we denote byu∞(x).
Moreover, thanks to the uniform bound (ii) and the differential equation (11),u belongs toW1,∞(R), and thus the sequenceuk is uniformly bounded (with respect to k) in W1,∞([0,1]). Thus u∞∈ W1,∞([0,1]), andu∞ is a continuous function.
According to Dini’s Theorem, we eventually deduce thatuk converges towardsu∞
in L∞([0,1]). Notice that u∞ is periodic by definition, and passing to the limit in equation (11), we deduce that u∞ is a solution of (11). Hence u∞ belongs to W1,∞(T) and satisfies
−u′′∞+ d
dx(A(x, u∞(x))) = 0,
which means exactly that u∞ is a periodic solution of equation (5); according to Proposition 2.1, there existsp∈Rsuch that u∞ =v(·, p). Eventually, sinceu∞ is a solution of (11), we infer thatα= ¯A(p).To sum up, we have proved that there existsp∈[p−, p+], such that ¯A(p) = ¯A(p±), and such that
k→∞lim sup
x∈[0,1]|u(x+k)−v(x, p)|= 0.
The above convergence is strictly equivalent to u(x)−v(x, p)→0 asx→ ∞,and thus the third point of the Proposition is proved. The limit asx→ −∞is treated similarly.
3.3. Further results on viscous shocks. We have gathered in this paragraph some results which will be important in the proof of Theorem 2.2. The first lemma gives a criterion allowing us to distinguish between the asymptotic states at±∞. Lemma 3.2. Let pl, pr∈Rsuch that A(p¯ l) = ¯A(pr), and let U be a shock profile such that
x→−∞lim [U(x)−v(x, pl)] = lim
x→+∞[U(x)−v(x, pr)] = 0.
Then
h∂vA(·, v(·, pl))i ≥0, h∂vA(·, v(·, pr))i ≤0.
Moreover, if one of the above inequalities is strict, then U converges exponentially fast toward the corresponding solution of equation (5); for instance, if
¯ ar:=
Z
T
∂vA(y, v(y, pr))dy <0,
then for all a∈(0,−a¯r), there exists a constantCa such that for all y∈[0,∞),
|U(y)−v(y, pr)| ≤Caexp(−ay).
Proof. SinceU is a shock profile andv(pl),v(pr) are solutions of equation (5), we have
U′(x) =A(x, U(x))−α,
∂xv(x, pl) =A(x, v(x, pl))−α,
∂xv(x, pr) =A(x, v(x, pr))−α,
whereαdenotes the common value of ¯A(pl) and ¯A(pr).
Consequently, the functionU−v(pr), for instance, satisfies the linear equation (16) ∂x(U(x)−v(x, pr)) =b(x)(U(x)−v(x, pr)),
where
b(x) = Z 1
0
∂vA(x, τ U(x) + (1−τ)v(x, pr))dτ.
Notice that sinceU converges towardsv(pr) asx→+∞, we obtain
(17) lim
x→+∞[b(x)−∂vA(x, v(x, pr))] = 0.
On the other hand, equation (16) implies that U(x)−v(x, pr) = [U(0)−v(0, pr)] exp
Z x 0
b(y)dy
. Once again, sinceU−v(pr) converges towards zero, we infer that
(18) lim
x→+∞
Z x 0
b(y)dy=−∞.
The first statement of the proposition follows easily from (17), (18); indeed, assume that ¯ar>0. Then there exists a positive numberK such that
x≥K⇒b(x)−∂vA(x, v(x, pr))≥ −a¯r
2 ,
and consequently, using the fact that x7→∂vA(x, v(x, pr)) is a periodic function, we obtain forx≥K
Z x K
b(y)dy ≥ Z x
K
∂vA(y, v(y, pr))dy−(x−K)¯ar
2
≥ ⌊x−K⌋¯ar−x¯ar
2 −C
≥ xa¯r
2 −C.
The above inequality is obviously in contradiction with (18). Hence ¯ar≤0, which proves the first statement in the proposition.
Now, assume that ¯ar<0, and choosea∈(0,−¯ar) arbitrary. As before, we pick K >0 such that
x≥K⇒b(x)−∂vA(x, v(x, pr))≤ −¯ar−a.
We then obtain an inequality of the type Z x
K
b(y)dy ≤ (−¯ar−a)(x−K) +⌊x−K⌋¯ar+C
≤ −ax+C.
Inserting this inequality back into the formula forU−v(pr) yields the exponential convergence result.
The next result is concerned with the integrability of the difference between two shock profiles.
Lemma 3.3. Let pl, pr∈Rsuch that pl6=pr andA(p¯ l) = ¯A(pr), and let U, V be two shock profiles with asymptotic states v(pl),v(pr).
Then U−V ∈L1(R).
Proof. Set
U0:=U(0), V0:=V(0),
and assume for instance that U0 ≤ V0. If U0 = V0, then U = V according to the Cauchy-Lipschitz Theorem (see the proof of Proposition 2.4), and the result is obvious. Thus we assume from now on thatU0< V0. As a consequence, we have
∀y∈R, v(y,min(pl, pr))< U(y)< V(y)< v(y,max(pl, pr)).
We recall that the sequence (U(k))k∈Z is monotonous, and converges towards v(0, pl) (resp. v(0, pr)) as k → −∞ (resp. k → +∞). Hence, there exists an integerk0∈Zsuch that
(19) U0< V0< U(k0), from which we infer thatU ≤V ≤τk0U.
As a consequence, it is sufficient to prove thatτkU−U is integrable, for allk∈Z. First, remember that τkU−U has a constant sign, since τkU and U are both shock profiles. Thus we only have to prove that the family
Z A
−A
(τkU−U)
remains bounded asA→ ∞. A simple calculation leads to Z A
−A
(τkU−U) = Z A
−A
U(y+k)dy− Z A
−A
U(y)dy
=
Z k+A k−A
U(y)dy− Z A
−A
U(y)dy
=
Z k+A A
U(y)dy− Z k−A
−A
U(y)dy.
Thus, recalling thatU is a bounded function, we obtain sup
A>0
Z A
−A
(τkU −U)
≤2kkUkL∞(R).
We deduce thatτkU−U ∈L1(R) for allk∈Z, and eventually thatU−V ∈L1(R) according to (19).
The next result is in fact the first part of the statement of Theorem 2.2:
Lemma 3.4. Letpl, pr∈Rsuch that the assumptions of Theorem 2.2 are satisfied, and letU be a viscous shock connecting v(pl) tov(pr).
Let u ∈ U +L1. Then there exists a unique shock profile V, with asymptotic statesv(pl)andv(pr), and such that
u∈V +L10(R).
Proof. According to Lemma 3.3, we already know that for every shock profile V, we haveu−V ∈L1. Hence, the question is to find a shock profileV such that (20)
Z
R
(u−V) = 0.
Notice that such a shock profile is necessarily unique: indeed, the Cauchy-Lipschitz uniqueness principle entails that the difference of two shock profiles is a function which keeps a constant sign. Hence, if V1, V2 are shock profiles satisfying R
R(V1− V2) = 0,thenV1=V2.
We now prove that there exists a shock profileV such that u−V ∈L10(R). As before, we set p− = min(pl, pr),p+ = max(pl, pr). For allξ ∈(v(0, p−), v(0, p+)), we denote byVξ the solution of
V′(x) =A(x, V(x))−A(p¯ l), V|x=0=ξ.
Then, according to Proposition 2.4 and Lemma 3.3, for allξ,Vξ is a shock profile connecting v(pl) to v(pr), and additionally u−Vξ ∈ L1(R). Moreover, if ξ > ξ′, thenVξ(x)> Vξ′(x) for allx; hence the function
F :ξ∈(v(0, p−), v(0, p+))7→
Z
R
(u(x)−Vξ(x))dx
is well-defined and decreasing with respect toξ; using classical results on differential equations, it can easily be proved that F is continuous. We wish to findξ0 such thatF(ξ0) = 0; thus it suffices to show that
ξ→v(0,plim−)+
F(ξ)>0 and lim
ξ→v(0,p+)−
F(ξ)<0.
The above result is a direct consequence of Lebesgue’s monotone convergence The- orem and of the fact that
(21) ∀x∈R, lim
ξ→v(0,p−)+
Vξ(x) =v(x, p−).
The same kind of result holds with v(p+). Indeed, letR >0 be arbitrary, and let ε >0. Without loss of generality, assume thatpr=p−. Then there existsK ∈N such that
x≥K⇒v(x, pr)≤U(x)≤v(x, pr) +ε.
In particular,τK+⌊R⌋+1U is a shock profile which satisfies τK+⌊R⌋+1U(x)≤v(x, pr) +ε ∀x∈[−R, R].
Let ¯ξ:=τK+⌊R⌋+1U(0) =U(K+⌊R⌋+ 1). The Cauchy-Lipschitz Theorem entails thatVξ¯=τK+⌊R⌋+1U. As a consequence, for allξ <ξ,¯ for allx∈[−R, R], we have
v(x, pr)≤Vξ(x)≤Vξ¯(x)≤v(x, pr) +ε.
The convergence result (21) follows, and thus there exists a shock profileV such thatu0∈V +L10(R).
The next lemma allows us to replace inequality (14) by an inequality in which the upper and lower bounds are shock profiles, which will be useful in the proof of Theorem 2.2 in Section 4.
Lemma 3.5. Letpl, prsuch that the hypotheses of Theorem 2.2 are satisfied. LetU be a shock profile connectingv(pl)tov(pr). Letu∈L∞(R)such thatu∈U+L10(R) and assume that for almost everyy∈R,
v(y,min(pr, pl))≤u(y)≤v(y,max(pl, pr)).
Let ε > 0 be arbitrary. Then there exists a function uε∈ U+L10(R), together with shock profiles U±ε connecting v(pl)tov(pr), such that
ku−uεkL1 ≤ε, U−ε ≤uε≤U+ε.
Proof. First, sinceu−U ∈L1(R), there exists a positive numberAε such that Z
|x|≥Aε|u−U| ≤ε.
Hence, for|x| ≥Aε, we takeuε(x) =U(x).
The definition ofuεon the interval [−Aε, Aε] is slighlty more technical, because of the various constraints bearing on uε. Once again, we assume that pl > pr in order to lighten the notation. We first consider a functionvε∈ C([−Aε, Aε]) which satisfies
Z
|x|≤Aε|u(x)−vε(x)|dx≤ε and such that
v(x, pr)< vε(x)< v(x, pl) ∀x∈[−Aε, Aε].
We denote byαε a positive number such that
v(x, pr) +αε≤vε(x)≤v(x, pl)−αε ∀x∈[−Aε, Aε].
Notice thatαε can be chosen as small as desired. For further purposes, we choose αεso that
αεAε≤2 Z
|x|≤Aε
(U −v(pr)).
The constraintuε∈U+L10(R) entails that the functionuεshould satisfy Z
|x|≤Aε
(uε−U) = 0.
However, the functionvεdoes not satisfy the above constraint in general: we merely have
Z
|x|≤Aε
(vε−U)
≤ Z
|x|≤Aε
(vε−u)
+ Z
|x|≤Aε
(u−U)
≤ Z
|x|≤Aε|vε−u|+ Z
|x|≥Aε|u−U|
≤ 2ε.
Assume for instance that R
|x|≤Aε(vε−U) > 0. We then define a non-negative functionρε∈L∞([−Aε, Aε]) such that
(22)
vε(x)−ρε(x)≥v(x, pr) +αε
2 a.e. on [−Aε, Aε] and
Z
|x|≤Aε
(vε−ρε−U) = 0.
Such a functionρεexists provided Z
|x|≤Aε
(vε−U)≤ Z
|x|≤Aε
vε−v(pr)−αε 2
, and the above inequality is equivalent to
Z
|x|≤Aε
(U−v(pr))≥αεAε 2 .
The previous condition is satisfied by definition ofαε. Thus there exists a function ρεwhich satisfies conditions (22).
We then set
uε(x) =vε(x)−ρε(x) forx∈[−Aε, Aε].
The construction is similar when R
|x|≤Aε(vε−U)<0.
At this stage, we have defined a functionuε∈U+L10which satisfies v(x, pr) +αε
2 < uε(x)≤v(x, pl)−αε
2 ∀x∈[−Aε, Aε], uε(x) =U(x) ∀x∈R\[−Aε, Aε],
and Z
R|u−uε| ≤4ε.
Now, by definition of the shock profileU, there exists a positive constantRεsuch that
x≥Rε⇒ |U(x)−v(x, pr)| ≤ αε 2 .
Letk+ be a positive integer such thatk+ > Rε+Aε.Then for allx∈[−Aε, Aε], we have
v(x, pr)≤τk+U(x)≤v(x, pr) +αε
2 ≤uε(x).
Similarly, there exists a negative integerk− such that for allx∈[−Aε, Aε], uε(x)≤v(x, pl)−αε
2 ≤τk−U(x).
Notice thatτk±U are also shock profiles. We now set
U+ε := sup(τk+U, U), U−ε := inf(τk−U, U).
Since shock profiles are ordered, the functionsU±ε are viscous shocks, and U−ε ≤uε≤U+ε a.e.
Hence the lemma is proved.
3.4. An application: the convex case. This paragraph is devoted to the anal- ysis of specific examples for which the existence of shock profiles and their stability can be proved.
Lemma 3.6. Assume that for all y ∈ T, A(y,·) is a convex function. Then the homogenized fluxA¯ is convex.
Furthermore, ifA(y,·)is strictly convex for all y, thenA¯ is also strictly convex, and thus satisfies the Oleinik condition of Corollary 2.1.
The convexity properties are proved in [15]. However, for the reader’s conve- nience, we have reproduced the proof in Appendix B. Oleinik’s condition is an immediate consequence of the strict convexity of the flux ¯A.
Example. Assume that the flux A is strictly convex in its second variable, and that the assumptions of Proposition 2.4 are satisfied. Then, with the same notation as in Proposition 2.4, we have
ql=p+ and qr=p−.
Indeed, according to Corollary 2.1, we have {ql, qr} ={p+, p−}. Moreover, since the fluxAis strictly convex,∂vA(y,·) is strictly increasing, and
∂vA(·, v(·, p−))
<
∂vA(·, v(·, p+)) . Proposition 3.2 then allows us to conclude thatp−=qr,p+=ql.
We now prove Corollary 2.2 (pending Proposition 2.6). Assume that the fluxA is given by
A(y, p) =V(y) +f(p),
withV andf satisfying the assumptions of Corollary 2.2. The existence of solutions of equation (5) follows immediately from Proposition 2.1; moreover, since the flux A is linear at infinity, hypothesis (7) is satisfied. As a consequence, for p > 0 sufficiently large, we have
A(y, v(y, p)) =V(y) +a+v(y, p) ∀y∈TN,
and thus ¯A(p) = hVi+a+p. Similarly, ¯A(p) = hVi −a−p for p < 0 with |p| sufficiently large. These formulas entail that ifα >0 is large enough, then, setting p± = ±(α− hVi)/a±, we have ¯A(p−) = ¯A(p+) = α. Since ¯A satisfies Oleinik’s condition, we deduce that there exists a shock profile connectingv(p−) andv(p+).
Additionally, if|p|is large enough, thenfis linear, say, on the intervals [infv(|p|)− 1,∞) and (−∞,supv(−|p|) + 1].Thus, for allξ∈[−1,1], we have
A(y, v(y, p) +ξ) =V(y) +f(v(y, p) +ξ) =A(y, v(y, p)) + sgn(p)asgn(p)ξ.
Hence the fluxAsatisfies the assumption of Proposition 2.6 for allplarge enough.
We infer that the solutions v(·, p±) are stable by the semi-group St under small perturbations in L10 which satisfy (42). Point (ii) in Corollary 2.2 then follows from Proposition 2.5 and the remark following it.
4. Stability of shock profiles in one space dimension - Part I This section is devoted to the proof of Theorem 2.2. Hence, throughout this section, we consider an initial data u0 which satisfies (14), and such that u0 ∈ U+L1, whereU is a stationary shock of equation (1). Using Lemma 3.4, we deduce that there exists another shock profile V such that u∈ V +L10(R). Then, using Lemma 3.5 together with the Contraction principle, we can restrict the analysis to the class of initial datau0such that
(23) ∃(U−, U+) shock profiles, U−≤u0≤U+.
Indeed, assume that Theorem 2.2 holds for allv0∈V+L10such that (23) is satisfied.
Consider now a functionu0 ∈V +L10 satisfying (14), and let ε >0 be arbitrary.
According to Lemma 3.5, there exists uε0 ∈ V +L10 satisfying (23) and such that ku0−uε0k1≤ε. The L1 contraction principle entails that for allt≥0,
kStu0−Vk1≤ kStu0−Stuε0k1+kStuε0−Vk1≤ε+kStuε0−Vk1.
Notice also that by the Contraction principle, the function t 7→ kStu0−Vk1 is non-increasing, and thus has a finite limit ast→ ∞. We infer that
∀ε >0, lim
t→∞kStu0−Vk1≤ε, and thusStu0 converges towardV as t→ ∞.
There remains to prove Theorem 2.2 for initial data which satisfy (23). As emphasized in Section 2, inequalities (14) or (23) should be seen as the analogues of (9) in the context of shock stability. The proof of Theorem 2.2 in this case relies on arguments from dynamical systems theory, which are due to S. Osher and J.
Ralston (see [17]; similar ideas are developed by D. Amadori and D. Serre in [1]).
The aim is to prove that the ω-limit set of the trajectoryStu0 is reduced to{V}, by using a suitable Lyapunov function. Hence, we first prove that the ω-limit set, denoted by Ω, is non-empty, then we state some properties of the ω-limit set, and eventually we prove that Ω ={V}.
First step. Compactness inL1of the trajectories.
Throughout this section, we set w(t) := Stu0−V. Notice first that by the comparison principle for equation (1), inequality (23) is preserved by the semi- groupSt: for allt≥0,we have
U−≤Stu0≤U+. Hence, for allt≥0,
U−−U ≤w(t)≤U+−U.
Since U+−U andU−−U are integrable functions, the family {w(t)}t≥0 is equi- integrable inL1. Moreover, sinceU+−U andU−U− are bounded, it follows that wis uniformly bounded inL∞. The functionwsatisfies a linear parabolic equation of the type
∂tw+∂y(b(t, y)w)−∂yyw= 0, t >0, y∈R,
with b ∈ L∞([0,∞)×R). Theorem 10.1 in Chapter III of [13] then implies that there existsα >0 such that for allt0≥1, for all R >0,
ku(t)kHα/2,α((t0,t0+1)×(−R,R)) <∞. Thus the family{w(t)}t≥0 is also equi-continuous inL1.
Whence it follows from the Riesz-Fr´echet-Kolmogorov Theorem that the family {w(t)}t≥0 is relatively compact inL1(R). Thus the ω-limit set
Ω :=n
W ∈V +L1(R),∃(tn)n∈N, tnn→∞−→ ∞, Stnu0→W inL1(R)o is non-empty.
Second step. Properties of theω-limit setΩ.
First, Ω is forward and backward invariant by the semi-groupSt, meaning that for allt≥0,
StΩ = Ω.
This important property is a generic one for ω-limit sets. It follows immediately, thanks to parabolic regularity, that all functions in Ω are smooth: Ω⊂Hloc1 (R),for instance. As a consequence, ifW ∈Ω andw1(t) :=StW, Theorem 6.1 in Chapter III of [13] entails thatw1∈L2([0, T], H2(BR))∩H1([0, T], L2(BR)) for allT, R >0.
The second property which is important for our analysis is the LaSalle invariance principle (see [14]), which requires the existence of a Lyapunov function. In the
case of scalar conservation laws, a classical choice for a Lyapunov function isF[u] = ku−Vk1. The Contraction principle entails that t 7→ F[Stu0] is non-increasing.
ThusF takes a constant value on Ω,which we denote byC0.
Eventually, using the conservation of mass, we deduce that Ω is a subset of V +L10.
Third step. Conclusion.
We now prove, using the parabolic structure of equation (24), that Ω ={V}. LetW0∈Ω be arbitrary, and letW(t) =St(W0). Notice thatW(t)∈Ω for all t≥0, according to the previous step. Moreover,W −V satisfies
∂t(W −V) +∂y(A(y, W)−A(y, V))−∂yy(W−V) = 0.
Multiplying the above equation by sgn(W−V), we obtain
∂t|W−V|+∂y[sgn(W −V) (A(y, W(t))−A(y, V))]−sgn(W−V)∂yy(W−V) = 0.
Letφbe a cut-off function, i.e. φ∈ C0∞(R), φ≥0 andφ≡1 in a neighbourhood of zero. For R >0, we setφR :=φ(·/R). We now multiply the above equality by φR and integrate on [t, t′]×R. Recalling that R
R|W(t)−V| = C0 for all t, we infer that for allt′ > t≥0, there exists a functionεt,t′ : [0,∞)→[0,∞) such that limR→∞εt,t′(R) = 0 and
Z t′ t
Z
R
sgn(W(s)−V)∂yy(W(s, y)−V(y))φR(y)ds dy
≤εt,t′(R).
Thus, using a slightly modified version of Lemma 1 in the Appendix, we infer that sgn(W(s)−V)∂yy(W(s)−V) =∂yy|(W(s)−V)|
almost everywhere and in the sense of distributions. Consequently, the function
|W −V|is a non-negative solution of a parabolic equation of the type
∂t|W −v|+∂y(b(t, y)|W −V|)−∂yy|W−V|= 0,
with b∈ L∞([0,∞R). We now conclude thanks to Harnack’s inequality (see [8]):
letx0∈Rbe arbitrary, and letKbe any compact set inRsuch thatx0∈K. Then there exists a constantCK such that
|(W0−V)(x0)| ≤ sup
x∈K|(W0−V)(x)| ≤CK inf
x∈K|(W|s=1−V)(x)|. Now, (W|s=1−V)∈L10∩Hloc1 (R), and thus there existsx1∈Rsuch that
W(1, x1)−V(x1) = 0.
Choose K such that x1 ∈ K. Then W0−V vanishes uniformly on K, and in particular, (W0−V)(x0) = 0. Since x0 was chosen arbitrarily, we deduce that W0=V. Hence Ω ={V},and Theorem 2.2 is proved.
5. Uniform in timea priori bounds for viscous scalar conservation laws
This section is devoted to the proof of Proposition 2.2. As far as possible, we will treat both models simultaneously. We set
w(t) :=Stu0−U0, t≥0.
The functionwsatisfies the following equation
(24) ∂tw(t, y) + divyB(y, w(t, y))−∆yw(t, y) = 0, t >0, y∈Q,
where
B(y, w) =A(y, U0(y) +w)−A(y, U0(y)), y∈Q, w∈R.
Due to the Contraction principle recalled in Section 2, it is known that w is bounded inL∞([0,∞), L1(Q)), and
(25) ∀t∈R+, kw(t)kL1 ≤ ku0−U0kL1.
The idea of this section is to use this uniform L1 bound in order to derive uni- form Lp bounds on w for all p ∈ [1,∞]. To that end, we proceed by induction on the exponent p. The first step is dedicated to the derivation of a differential inequality relating the derivative of theLp norm to a viscous dissipation term. The calculations are very similar to those developed in [5] to derivea priori bounds for solutions of equation (5). Then, we use Poincar´e inequalities to control theLpnorm by the dissipation. Eventually, we conclude thanks to a Gronwall type argument.
Preliminary for the whole space case.
We begin by recalling some regularity results about the solutions of equation (1) in the case Q=R. According to the papers by Kruˇzkov [11, 12], it is kown that w∈L∞loc([0,∞), L∞(Q)). As a consequence,w∈L∞loc([0,∞), Lp(Q)) for allp.
Then, multiplying (24) by wχ where χ ∈ C0∞(R) is an arbitrary non-negative cut-off function, and integrating in space and time, it is easily proved that for all T >0,wsatisfies an inequality of the type
Z T 0
Z
R
|∂yw(s, y)|2χ(y)dy ds≤CT,
where the constantCT depends on T,kwkL∞([0,T]×R)and kwt=0k1, but not onχ.
We deduce that∂yw∈L2loc([0,∞), L2(R)).
First step. A differential inequality.
In this step, we treat the periodic and the full space models simultaneously; our goal is to prove an inequality of the type
d dt
Z
|w|q+1+cq
Z
∇|w|q+12
2
≤Cq
Z
|w|q+n+ Z
|w|q+1
,
whereq≥1 is arbitrary,nis the exponent appearing in (8), and the constantscq
andCq depend onq,n,N, andkU0kW1,∞.
To that end, we takeq ≥1, multiply (24) by w|w|q−1 and integrate on Q; we obtain
1 q+ 1
d dt
Z
Q|w|q+1+q Z
Q|∇w|2|w|q−1=
=q Z
Q∇yw(t, y)·B(y, w(t, y))|w(t, y)|q−1dy.
Notice that all terms are well-defined thanks to the preliminary step.
For (y, w)∈Q×R, set
bq(y, w) =q Z w
0
B(y, w′)|w′|q−1dw′;
then
−q Z
T
∇yw(t, y)·B(y, w(t, y))|w(t, y)|q−1dy
= Z
T
[−divy(bq(y, w(t, y))) + (divybq)(y, w(t, y))]dy
= q
Z
T
Z w(t,y) 0
(divyB)(y, w′)|w′|q−1dw′. Thus, we now compute, for (y, w′)∈Q×R,
divyB(y, w′) = divy[A(y, U0(y) +w′)−A(y, U0(y))]
= (divyA)(y, U0(y) +w′)−(divyA)(y, U0(y)) + ∇yU0·[(∂vA)(y, U0(y) +w′)−(∂vA)(y, U0(y))]. Consequently, according to hypothesis (8), we deduce that there exists a positive constantC depending only onkU0kW1,∞ andqsuch that
q
Z
∇yw(t, y)·B(y, w(t, y))|w(t, y)|q−1dy ≤C
Z
|w(t)|q+1+ Z
|w(t)|q+n
. Eventually, we infer that for all q ≥1, there exist positive constants cq, Cq such that for allt >0,
(26) d dt
Z
|w(t)|q+1+cq
Z
∇|w(t)|q+12
2
≤Cq
Z
|w(t)|q+1+ Z
|w(t)|q+n
.
Second step. Control ofLp norms by the dissipation term (Poincar´e inequalities).
In this step, we treat the periodic case and the whole space case separately, and we begin with the periodic case.
First, remember that for all p ∈ (1,∞) such that p1 ≥ 12 − N1, there exists a positive constantCp such that for allφ∈Hper1 (TN),
(27) kφ− hφikp≤Cpk∇φk2. Takingφ=|w|q+12 , we deduce that
kwkr≤Cr
∇|w|q+12
2 q+1
2 +kwkq+12
, wherer∈(1,∞) is such that
(28) 1
r ≥ 1
q+ 1 − 2 N(q+ 1).
Now, the idea is to interpolate theLn+q and the Lq+1 norms in the right-hand side of inequality (26) between L1 and Lr, where r satisfies the constraint above.
It can be easily checked that whenn < N+ 2)/N, we have 1
n+q > 1
q+ 1− 2 N(q+ 1); hence the interpolation is always possible, and we have kwkq+1≤ kwk1−α1 kwkαr, kwkq+n≤ kwk1−β1 kwkβr,
